Structure theorem for finitely generated modules over domains of principal ideals

The structural theorem for finitely generated modules over domains of principal ideals is a generalization of the classification theorem for finitely generated abelian groups . This theorem provides a general way of understanding some results on the canonical forms of matrices.

Theorem

If a vector space over a field k has a finite generating set, one can always choose a basis from it, so that the vector space is isomorphic to k ⁿ . For finitely generated modules, this is already not true (counterexample - $\mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2}}$ $\mathbb Z_2$ , which is generated by one element as a Z- module), but such a module can be represented as a factor module of the form R ⁿ / A (to see this, it is enough to map the basis R ⁿ into a generating set and use the homomorphism theorem ). Changing the choice of a basis in R ⁿ and a generating set in the module, we can bring this factor to a simple form, and this gives a structural theorem.

The statement of the structural theorem is usually given in two different forms.

Invariant Decomposition

Each finitely generated module M over a region of principal ideals R is isomorphic to a unique module of the form

\bigoplus _{i}R/(d_{i})=R/(d_{1})\oplus R/(d_{2})\oplus \cdots \oplus R/(d_{n}),

{\ displaystyle \ bigoplus _ {i} R / (d_ {i}) = R / (d_ {1}) \ oplus R / (d_ {2}) \ oplus \ cdots \ oplus R / (d_ {n}) ,}

\bigoplus _{i}R/(d_{i})=R/(d_{1})\oplus R/(d_{2})\oplus \cdots \oplus R/(d_{n}),

Where $(d_{i})\neq R$ ${\ displaystyle (d_ {i}) \ neq R}$ $(d_i) \neq R$ and $d_{i}\vert d_{i+1}$ ${\ displaystyle d_ {i} \ vert d_ {i + 1}}$ $d_i \vert d_{i+1}$ (i.e $d_{i+1}$ ${\ displaystyle d_ {i + 1}}$ $d_{i+1}$ divided by $d_{i}$ ${\ displaystyle d_ {i}}$ $d_{i}$ ) Nonzero order $(d_{i})$ ${\ displaystyle (d_ {i})}$ $(d_i)$ defined uniquely, like a number $(d_{i})=0$ ${\ displaystyle (d_ {i}) = 0}$ $(d_i)=0$ .

Thus, to indicate a finitely generated module M, it is sufficient to indicate nonzero $(d_{i})$ ${\ displaystyle (d_ {i})}$ $(d_i)$ (satisfying two conditions) and the number equal to zero $(d_{i})$ ${\ displaystyle (d_ {i})}$ $(d_i)$ . Items $d_{i}$ ${\ displaystyle d_ {i}}$ $d_{i}$ are uniquely determined up to multiplication by invertible elements of the ring and are called invariant factors.

Primary factorization

Each finitely generated module M over a region of principal ideals R is isomorphic to a unique module of the form

\bigoplus _{i}R/(q_{i}),

{\ displaystyle \ bigoplus _ {i} R / (q_ {i}),}

Where $(q_{i})\neq R$ ${\ displaystyle (q_ {i}) \ neq R}$ and all $(q_{i})$ ${\ displaystyle (q_ {i})}$ - primary ideals . At the same time $q_{i}$ ${\ displaystyle q_ {i}}$ defined uniquely (up to multiplication by invertible elements).

In the case when the ring R is Euclidean , all primary ideals are degrees of simple ones , i.e. $(q_{i})=(p_{i}^{r_{i}})$ ${\ displaystyle (q_ {i}) = (p_ {i} ^ {r_ {i}})}$ .

Sketch of evidence for Euclidean rings

Many areas of the main ideals are also Euclidean rings . Moreover, the proof for Euclidean rings is somewhat simpler; here are his main steps.

Lemma. Let A be a Euclidean ring, M be a free A -module, and N be its submodule. Then N is also free, its rank does not exceed the rank of M , and there exists a basis {e ₁ , e ₂ , ... e _m } of the module M and such nonzero elements {u ₁ , ... u _k } of the ring A such that {u ₁ e ₁ , ... u _k e _k } - the basis of N and u _{i + 1} is divided by u _i .

The proof that N is free is carried out by induction on m . The base m = 0 is obvious; we prove the induction step. Let M _{1 be} generated by the elements {e ₁ , ... e _m-1 }, N ₁ is the intersection of M ₁ and N is free by the assumption of induction. The last coordinates of the elements N in the basis {e ₁ , ... e _m } form a submodule of the ring A (that is, an ideal), A is the ring of principal ideals, therefore this ideal is generated by one element; if the ideal is zero - N coincides with N ₁ , if it is generated by the element k , it is enough to add one vector to the basis of N ₁ , the last coordinate of which is k .

Now we can write a matrix with elements from A corresponding to the embedding of N in M : in the columns of the matrix, we write the coordinates of the basis vectors N in some basis M. Let us describe an algorithm for reducing this matrix to a diagonal form by elementary transformations . By interchanging the rows and columns, we move the nonzero element a with the lowest norm to the upper left corner. If all elements of the matrix are divided by it, we subtract the first row from the others with such a coefficient that all elements of the first column (except the first element) become zero; then in the same way we subtract the first column and go to the transformations of the square remaining in the lower right corner, the dimension of which is one less. If there is an element b not divisible by a - we can reduce the minimum norm for nonzero matrix elements by applying the Euclidean algorithm to the pair ( a , b ) (elementary transformations allow this). Since the norm is a natural number, sooner or later we will come to a situation where all elements of the matrix are divided by a . It is easy to see that at the end of this algorithm, the bases M and N satisfy all the conditions of the lemma.

The end of the proof. Consider a finitely generated module T with a system of generators {e ₁ , ... e _m }. There is a homomorphism from a free module $A^{m}$ ${\ displaystyle A ^ {m}}$ into this module displaying the basis $A^{m}$ ${\ displaystyle A ^ {m}}$ into the system of generators. Applying the homomorphism theorem to this map, we find that T is isomorphic to the factor $A^{m}/{\text{ker}}f$ ${\ displaystyle A ^ {m} / {\ text {ker}} f}$ . We give the bases $A^{m}$ ${\ displaystyle A ^ {m}}$ and ${\text{ker}}f$ ${\ displaystyle {\ text {ker}} f}$ to the type of bases in the lemma. Easy to see that

A^{m}/(u_{1}e_{1},u_{2}e_{2},\ldots u_{k}e_{k})\cong A/(u_{1})\oplus \ldots \oplus A/(u_{k})\oplus A^{n-k}

{\ displaystyle A ^ {m} / (u_ {1} e_ {1}, u_ {2} e_ {2}, \ ldots u_ {k} e_ {k}) \ cong A / (u_ {1}) \ oplus \ ldots \ oplus A / (u_ {k}) \ oplus A ^ {nk}}

Each finite term here can be decomposed into a product of primary, since the ring A is factorial (see the article on the Chinese remainder theorem ). To prove the uniqueness of this decomposition, we need to consider the torsion submodule (then the dimension of the free part is described in invariant terms as the dimension of the torsion factor), as well as the p- torsion submodule for each simple element p of the ring A. The number of terms of the form $A/(p^{n})$ ${\ displaystyle A / (p ^ {n})}$ (for all n ) is invariantly described as the dimension of the submodule of elements canceled by multiplication by p , as a vector space over a field $A/(p)$ ${\ displaystyle A / (p)}$ .

Consequences

Happening $R=\mathbb {Z}$ ${\ displaystyle R = \ mathbb {Z}}$ gives a classification of finitely generated abelian groups .

Let T be a linear operator on a finite-dimensional vector space V over a field K. V can be considered as a module over $K[T]$ ${\ displaystyle K [T]}$ (indeed, its elements can be multiplied by scalars and by T ), finite-dimensionality implies finitely generated and the absence of a free part. The last invariant factor is the minimal polynomial , and the product of all invariant factors is the characteristic polynomial . Choosing the standard form of the matrix of the operator T acting on the space $K[T]/p(T)$ ${\ displaystyle K [T] / p (T)}$ , we obtain the following forms of the matrix T on the space V :

invariant factors + accompanying matrix gives the Frobenius normal form
primary factors + Jordan cell gives the Jordan normal form (in the case when the field K is algebraically closed ).

Notes

Bourbaki N. Algebra. Part 3. Modules, rings, forms. - M.: Science, 1966. Chapter VII.
Vinberg E. B. , Algebra course. - M .: Publishing house Factorial Press, 2001.
P. Aluffi. Algebra: Chapter 0 (Graduate Studies in Mathematics) - American Mathematical Society, 2009 - ISBN 0-82184-781-3 .
Hungerford, Thomas W. (1980), Algebra , New York: Springer, p. 218–226, Section IV.6: Modules over a Principal Ideal Domain, ISBN 978-0-387-90518-1